Numerical Optimization Techniques

Notification.- 1. An Introduction to Optimization Theory.- 1. Convex Sets and Convex Functions.- 2. Differentiability of Convex Functions.- 3. Necessary and Sufficient Conditions of a Local Extremum of Functions of Many Variables.- 4. Necessary and Sufficient Conditions for a Minimum of Functions on Sets.- 5. Properties of Minimax Problems.- 6. Conditions for a Minimum in Nonlinear Programming Problems Without Differentiability.- 7. Conditions for a Minimum in Nonlinear Programming Problems With Differentiability.- 8. Necessary Conditions for a Minimum in Optimal Control Problems.- 2. Convergence Theorems and Their Application to the Investigation of Numerical Methods.- 1. Stability of the First-Order Approximation.- 2. The Method of Lyapunov Functions.- 3. Theorems on Convergence of Iterative Processes.- 4. Convergence of Processes Generated by Multivalued Mappings.- 5. Methods for Solving Systems of Nonlinear Equations.- 6. Numerical Methods for Finding a Minimax.- 3. The Penalty-Function Method.- 1. The Exterior Penalty-Function Method.- 2. Estimation of Accuracy.- 3. The Cost-Function Parametrization Method.- 4. The Interior Penalty-Function Method.- 5. The Linearization Method.- 4. Numerical Methods for Solving Nonlinear Programming Problems Using Modified Lagrangians.- 1. The Simplest Modification of the Lagrangian.- 2. Modified Lagrangians.- 3. Proof of Convergence for the Simple Iteration Method.- 4. Solution of Convex Programming Problems.- 5. Reduction to a Maximin Problem.- 6. Reduction to a Minimax Problem.- 5. Relaxation Methods for Solving Nonlinear Programming Problems.- 1. Application of the Reduced Gradient Method to Solving Problems With Equality-Type Constraints.- 2. A Generalization of the Reduced Gradient Method.- 3. A Discrete Version of the Reduced Gradient Method.- 4. The Conditional Gradient Method.- 5. The Gradient Projection Method.- 6. Numerical Methods for Solving Optimal Control Problems.- 1. Basic Computational Formulas.- 2. Necessary